Combinational Functions and Circuits Soon Tee Teoh

Decoders From truth table, circuit for 2x4 decoder is: Note: Each output is a 2-variable minterm (X'Y', X'Y, XY' or XY) F0 = X'Y' F1 = X'Y F2 = XY' F3 = XY X Y From Figure 3-18, page 122

Decoders Design a 3x8 decoder. F1 = x'y'z x z y F0 = x'y'z' F2 = x'yz'

Decoders Convert binary information from n input lines to 2n output lines. Known as n-to-m-line decoder (m = 2n). May be used to generate the 2n minterms of n input variables.

Decoders In general, for an n-bit code, a decoder could select up to 2n lines: : n-bit code n to 2n decoder 2n output lines

Decoder Expansion Build a decoder with n+1 inputs using n-input decoder. A2A1A0 000 010 001 011 100 110 101 111 A0 A1 From Figure 3-19, page 122 A2

Decoders: Implementing Functions A Boolean function, in sum-of-minterms form, can be implemented using: a decoder to generate the minterms, and an OR gate to form the sum. Any combinational circuit with n inputs and m outputs can be implemented with an n:2n decoder with m OR gates. Good when circuit has many outputs, and each function is expressed with few minterms.

Decoders: Implementing Functions Example: Full adder S(x, y, z) =  m(1,2,4,7) C(x, y, z) =  m(3,5,6,7) 3x8 Dec S2 S1 S0 x y z 1 2 3 4 5 6 7 S C From Figure 3-22, page 127

Decoders: Implementing Functions F1 = A’B’CD + A’BC’D + ABCD Exercise: What are F2 and F3 ?

Multiplexer Selects from one of many input lines and directs it to a single output line. Selection is controlled by selection inputs. For a 2n-to-1 multiplexer, there are 2n data input lines and n selection lines whose bit combination determines which input is selected. 2n Data Inputs Data Output n Selection Inputs

Making a Multiplexer using a Decoder 2-to-4 decoder I0 S0 I1 Y S1 I2 I3 What is Y when S0 is 1 and S1 is 0?

Selecting m-bit data Until now, we have examined single-bit data selected by a MUX. What if we want to select m-bit data/words? Combine MUX blocks in parallel with common select and enable signals

Implementing Combinational Function using Multiplexer For function with n variables, we need n-1 selection inputs for mux Example: F(X,Y,Z) = X’Y’Z + X’YZ’ + XYZ’ + XYZ There are n=3 inputs, thus we need a 2-to-1 MUX The first n-1 (=2) variables serve as the selection lines The remaining variable goes into input lines (See pp. 136-139 of your text)

Example Z Y X F 0 F=0 1 1 F=1 1 F= X´ 0 F= X 1 X´ X F Z Y

Example Y X Z F 0 F=Z 1 0 F=0 1 F= Z´ 1 F= 1 Z Z´ 1 F Y X